Fit log returns to F-S skew standardized Student-t distribution.
m is the location parameter.
s is the scale parameter.
nu is the estimated shape parameter (degrees of freedom).
xi is the estimated skewness parameter.

Returns data 2011-2023.

For 2011, medium risk data is used in the high risk data set, as no high risk fund data is available prior to 2012.
vmrl is a long version of Velliv medium risk data, from 2007 to 2023. For 2007 to 2011 (both included) no high risk data is available.

PFA medium risk is risk profile B.
PFA high risk is risk profile D.

Summary of log-returns

The summary statistics are transformed back to the scale of gross returns by taking \(exp()\) of each summary statistic. (Note: Taking arithmetic mean of gross returns directly is no good. Must be geometric mean.)

vmr vhr vmrl pmr phr mmr mhr vmr_phr vhr_pmr
Min. : 0.868 0.849 0.801 0.904 0.878 0.885 0.864 0.874 0.873
1st Qu.: 1.044 1.039 1.013 1.042 1.068 1.059 1.061 1.064 1.055
Median : 1.097 1.099 1.085 1.084 1.128 1.089 1.127 1.119 1.104
Mean : 1.067 1.080 1.057 1.063 1.089 1.065 1.085 1.079 1.072
3rd Qu.: 1.136 1.160 1.128 1.107 1.182 1.121 1.144 1.139 1.134
Max. : 1.168 1.214 1.193 1.141 1.208 1.143 1.211 1.183 1.163

Ranking

Min. : ranking 1st Qu.: ranking Median : ranking Mean : ranking 3rd Qu.: ranking Max. : ranking
0.904 pmr 1.068 phr 1.128 phr 1.089 phr 1.182 phr 1.214 vhr
0.885 mmr 1.064 vmr_phr 1.127 mhr 1.085 mhr 1.160 vhr 1.211 mhr
0.878 phr 1.061 mhr 1.119 vmr_phr 1.080 vhr 1.144 mhr 1.208 phr
0.874 vmr_phr 1.059 mmr 1.104 vhr_pmr 1.079 vmr_phr 1.139 vmr_phr 1.193 vmrl
0.873 vhr_pmr 1.055 vhr_pmr 1.099 vhr 1.072 vhr_pmr 1.136 vmr 1.183 vmr_phr
0.868 vmr 1.044 vmr 1.097 vmr 1.067 vmr 1.134 vhr_pmr 1.168 vmr
0.864 mhr 1.042 pmr 1.089 mmr 1.065 mmr 1.128 vmrl 1.163 vhr_pmr
0.849 vhr 1.039 vhr 1.085 vmrl 1.063 pmr 1.121 mmr 1.143 mmr
0.801 vmrl 1.013 vmrl 1.084 pmr 1.057 vmrl 1.107 pmr 1.141 pmr

Correlations and covariance

Correlations

vmr vhr pmr phr
vmr 1.000 0.993 0.938 0.941
vhr 0.993 1.000 0.917 0.939
pmr 0.938 0.917 1.000 0.957
phr 0.941 0.939 0.957 1.000

Covariances

vmr vhr pmr phr
vmr 0.007 0.009 0.005 0.008
vhr 0.009 0.011 0.006 0.010
pmr 0.005 0.006 0.004 0.007
phr 0.008 0.010 0.007 0.011

Compare pension plans

Risk of loss

Risk of loss at least as big as row name in percent for a single period (year).

Skewed \(t\)-distribution (sstd):

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
0 17.167 19.667 11.833 16.000 16.667 17.000 15.000 14.500
5 9.167 12.500 5.667 9.333 8.500 10.500 8.667 7.333
10 5.000 8.000 3.000 5.333 4.500 6.500 5.000 3.833
25 0.667 2.167 0.500 0.833 0.667 1.667 1.000 0.333
50 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
90 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
99 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

Standardized \(t\)-distribution (std):

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
0 17.333 20.333 8.833 26.667 7.167 26.500 26.667 8.833
5 7.667 10.333 4.333 14.500 3.667 14.167 13.333 5.000
10 3.000 4.667 2.333 6.333 2.000 6.000 5.167 2.833
25 0.000 0.000 0.333 0.000 0.333 0.000 0.000 0.667
50 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
90 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
99 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

Normal distribution:

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
0 21.167 21.667 16.500 19.667 18.667 20.167 19.833 19.167
5 7.333 9.500 3.333 8.500 5.167 8.667 7.667 6.333
10 1.500 2.833 0.000 2.667 0.500 2.500 1.833 1.167
25 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
50 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
90 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
99 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

Worst ranking for loss percentiles

Skewed \(t\)-distribution (sstd):

0 ranking 5 ranking 10 ranking 25 ranking 50 ranking 90 ranking 99 ranking
19.667 vhr 12.500 vhr 8.000 vhr 2.167 vhr 0 vmr 0 vmr 0 vmr
17.167 vmr 10.500 mhr 6.500 mhr 1.667 mhr 0 vhr 0 vhr 0 vhr
17.000 mhr 9.333 phr 5.333 phr 1.000 vmr_phr 0 pmr 0 pmr 0 pmr
16.667 mmr 9.167 vmr 5.000 vmr 0.833 phr 0 phr 0 phr 0 phr
16.000 phr 8.667 vmr_phr 5.000 vmr_phr 0.667 vmr 0 mmr 0 mmr 0 mmr
15.000 vmr_phr 8.500 mmr 4.500 mmr 0.667 mmr 0 mhr 0 mhr 0 mhr
14.500 vhr_pmr 7.333 vhr_pmr 3.833 vhr_pmr 0.500 pmr 0 vmr_phr 0 vmr_phr 0 vmr_phr
11.833 pmr 5.667 pmr 3.000 pmr 0.333 vhr_pmr 0 vhr_pmr 0 vhr_pmr 0 vhr_pmr

Standardized \(t\)-distribution (std):

0 ranking 5 ranking 10 ranking 25 ranking 50 ranking 90 ranking 99 ranking
26.667 phr 14.500 phr 6.333 phr 0.667 vhr_pmr 0 vmr 0 vmr 0 vmr
26.667 vmr_phr 14.167 mhr 6.000 mhr 0.333 pmr 0 vhr 0 vhr 0 vhr
26.500 mhr 13.333 vmr_phr 5.167 vmr_phr 0.333 mmr 0 pmr 0 pmr 0 pmr
20.333 vhr 10.333 vhr 4.667 vhr 0.000 vmr 0 phr 0 phr 0 phr
17.333 vmr 7.667 vmr 3.000 vmr 0.000 vhr 0 mmr 0 mmr 0 mmr
8.833 pmr 5.000 vhr_pmr 2.833 vhr_pmr 0.000 phr 0 mhr 0 mhr 0 mhr
8.833 vhr_pmr 4.333 pmr 2.333 pmr 0.000 mhr 0 vmr_phr 0 vmr_phr 0 vmr_phr
7.167 mmr 3.667 mmr 2.000 mmr 0.000 vmr_phr 0 vhr_pmr 0 vhr_pmr 0 vhr_pmr

Normal distribution:

0 ranking 5 ranking 10 ranking 25 ranking 50 ranking 90 ranking 99 ranking
21.667 vhr 9.500 vhr 2.833 vhr 0 vmr 0 vmr 0 vmr 0 vmr
21.167 vmr 8.667 mhr 2.667 phr 0 vhr 0 vhr 0 vhr 0 vhr
20.167 mhr 8.500 phr 2.500 mhr 0 pmr 0 pmr 0 pmr 0 pmr
19.833 vmr_phr 7.667 vmr_phr 1.833 vmr_phr 0 phr 0 phr 0 phr 0 phr
19.667 phr 7.333 vmr 1.500 vmr 0 mmr 0 mmr 0 mmr 0 mmr
19.167 vhr_pmr 6.333 vhr_pmr 1.167 vhr_pmr 0 mhr 0 mhr 0 mhr 0 mhr
18.667 mmr 5.167 mmr 0.500 mmr 0 vmr_phr 0 vmr_phr 0 vmr_phr 0 vmr_phr
16.500 pmr 3.333 pmr 0.000 pmr 0 vhr_pmr 0 vhr_pmr 0 vhr_pmr 0 vhr_pmr

Chance of min gains

Chance of gains of at least x percent for a single period (year).
x values are row names.

Skewed \(t\)-distribution (sstd):

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
0 82.833 80.333 88.167 84.000 83.333 83.000 85.000 85.500
5 68.333 69.333 71.667 73.000 66.500 72.500 73.167 71.167
10 44.667 53.333 32.500 55.833 35.667 56.167 53.000 46.000
25 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
50 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
100 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

Standardized \(t\)-distribution (std):

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
0 82.667 79.667 91.167 73.333 92.833 73.500 73.333 91.167
5 65.833 65.000 80.000 58.167 84.000 58.000 56.667 82.667
10 44.500 48.000 54.833 42.500 62.333 42.167 39.500 65.500
25 7.000 11.667 6.667 10.000 7.500 9.500 7.000 12.167
50 0.167 0.500 0.833 0.000 1.000 0.000 0.000 1.833
100 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.167

Normal distribution:

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
0 78.833 78.333 83.500 80.333 81.333 79.833 80.167 80.833
5 57.667 61.333 57.667 64.167 57.833 63.000 61.833 60.167
10 35.167 42.500 29.000 46.167 32.167 44.333 41.333 37.167
25 2.167 6.667 0.000 8.333 0.833 7.167 4.833 2.333
50 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
100 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

Best ranking for gains percentiles

Skewed \(t\)-distribution (sstd):

0 ranking 5 ranking 10 ranking 25 ranking 50 ranking 100 ranking
88.167 pmr 73.167 vmr_phr 56.167 mhr 0 vmr 0 vmr 0 vmr
85.500 vhr_pmr 73.000 phr 55.833 phr 0 vhr 0 vhr 0 vhr
85.000 vmr_phr 72.500 mhr 53.333 vhr 0 pmr 0 pmr 0 pmr
84.000 phr 71.667 pmr 53.000 vmr_phr 0 phr 0 phr 0 phr
83.333 mmr 71.167 vhr_pmr 46.000 vhr_pmr 0 mmr 0 mmr 0 mmr
83.000 mhr 69.333 vhr 44.667 vmr 0 mhr 0 mhr 0 mhr
82.833 vmr 68.333 vmr 35.667 mmr 0 vmr_phr 0 vmr_phr 0 vmr_phr
80.333 vhr 66.500 mmr 32.500 pmr 0 vhr_pmr 0 vhr_pmr 0 vhr_pmr

Standardized \(t\)-distribution (std):

0 ranking 5 ranking 10 ranking 25 ranking 50 ranking 100 ranking
92.833 mmr 84.000 mmr 65.500 vhr_pmr 12.167 vhr_pmr 1.833 vhr_pmr 0.167 vhr_pmr
91.167 pmr 82.667 vhr_pmr 62.333 mmr 11.667 vhr 1.000 mmr 0.000 vmr
91.167 vhr_pmr 80.000 pmr 54.833 pmr 10.000 phr 0.833 pmr 0.000 vhr
82.667 vmr 65.833 vmr 48.000 vhr 9.500 mhr 0.500 vhr 0.000 pmr
79.667 vhr 65.000 vhr 44.500 vmr 7.500 mmr 0.167 vmr 0.000 phr
73.500 mhr 58.167 phr 42.500 phr 7.000 vmr 0.000 phr 0.000 mmr
73.333 phr 58.000 mhr 42.167 mhr 7.000 vmr_phr 0.000 mhr 0.000 mhr
73.333 vmr_phr 56.667 vmr_phr 39.500 vmr_phr 6.667 pmr 0.000 vmr_phr 0.000 vmr_phr

Normal distribution:

0 ranking 5 ranking 10 ranking 25 ranking 50 ranking 100 ranking
83.500 pmr 64.167 phr 46.167 phr 8.333 phr 0 vmr 0 vmr
81.333 mmr 63.000 mhr 44.333 mhr 7.167 mhr 0 vhr 0 vhr
80.833 vhr_pmr 61.833 vmr_phr 42.500 vhr 6.667 vhr 0 pmr 0 pmr
80.333 phr 61.333 vhr 41.333 vmr_phr 4.833 vmr_phr 0 phr 0 phr
80.167 vmr_phr 60.167 vhr_pmr 37.167 vhr_pmr 2.333 vhr_pmr 0 mmr 0 mmr
79.833 mhr 57.833 mmr 35.167 vmr 2.167 vmr 0 mhr 0 mhr
78.833 vmr 57.667 vmr 32.167 mmr 0.833 mmr 0 vmr_phr 0 vmr_phr
78.333 vhr 57.667 pmr 29.000 pmr 0.000 pmr 0 vhr_pmr 0 vhr_pmr

MC risk percentiles

Risk of loss at least as big as row name in percent from first to last period.

Skewed \(t\)-distribution (sstd):

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
0 1.46 4.12 1.91 0.82 0.10 0.15 0.05 0.28
5 1.22 3.69 1.71 0.71 0.08 0.12 0.02 0.24
10 1.01 3.27 1.56 0.63 0.08 0.08 0.02 0.22
25 0.67 2.32 1.13 0.35 0.02 0.03 0.00 0.11
50 0.18 1.10 0.62 0.10 0.00 0.01 0.00 0.02
90 0.02 0.11 0.16 0.01 0.00 0.00 0.00 0.00
99 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

Standardized \(t\)-distribution (std):

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
0 0.08 0.07 0.71 0.43 0 0 0 0.02
5 0.07 0.05 0.65 0.34 0 0 0 0.01
10 0.06 0.03 0.60 0.22 0 0 0 0.01
25 0.03 0.00 0.50 0.07 0 0 0 0.00
50 0.00 0.00 0.32 0.00 0 0 0 0.00
90 0.00 0.00 0.11 0.00 0 0 0 0.00
99 0.00 0.00 0.00 0.00 0 0 0 0.00

Normal distribution:

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
0 0.02 0.03 0.01 0 0 0 0 0
5 0.01 0.02 0.00 0 0 0 0 0
10 0.00 0.02 0.00 0 0 0 0 0
25 0.00 0.01 0.00 0 0 0 0 0
50 0.00 0.00 0.00 0 0 0 0 0
90 0.00 0.00 0.00 0 0 0 0 0
99 0.00 0.00 0.00 0 0 0 0 0

Worst ranking for MC loss percentiles

Skewed \(t\)-distribution (sstd):

0 ranking 5 ranking 10 ranking 25 ranking 50 ranking 90 ranking 99 ranking
4.12 vhr 3.69 vhr 3.27 vhr 2.32 vhr 1.10 vhr 0.16 pmr 0 vmr
1.91 pmr 1.71 pmr 1.56 pmr 1.13 pmr 0.62 pmr 0.11 vhr 0 vhr
1.46 vmr 1.22 vmr 1.01 vmr 0.67 vmr 0.18 vmr 0.02 vmr 0 pmr
0.82 phr 0.71 phr 0.63 phr 0.35 phr 0.10 phr 0.01 phr 0 phr
0.28 vhr_pmr 0.24 vhr_pmr 0.22 vhr_pmr 0.11 vhr_pmr 0.02 vhr_pmr 0.00 mmr 0 mmr
0.15 mhr 0.12 mhr 0.08 mmr 0.03 mhr 0.01 mhr 0.00 mhr 0 mhr
0.10 mmr 0.08 mmr 0.08 mhr 0.02 mmr 0.00 mmr 0.00 vmr_phr 0 vmr_phr
0.05 vmr_phr 0.02 vmr_phr 0.02 vmr_phr 0.00 vmr_phr 0.00 vmr_phr 0.00 vhr_pmr 0 vhr_pmr

Standardized \(t\)-distribution (std):

0 ranking 5 ranking 10 ranking 25 ranking 50 ranking 90 ranking 99 ranking
0.71 pmr 0.65 pmr 0.60 pmr 0.50 pmr 0.32 pmr 0.11 pmr 0 vmr
0.43 phr 0.34 phr 0.22 phr 0.07 phr 0.00 vmr 0.00 vmr 0 vhr
0.08 vmr 0.07 vmr 0.06 vmr 0.03 vmr 0.00 vhr 0.00 vhr 0 pmr
0.07 vhr 0.05 vhr 0.03 vhr 0.00 vhr 0.00 phr 0.00 phr 0 phr
0.02 vhr_pmr 0.01 vhr_pmr 0.01 vhr_pmr 0.00 mmr 0.00 mmr 0.00 mmr 0 mmr
0.00 mmr 0.00 mmr 0.00 mmr 0.00 mhr 0.00 mhr 0.00 mhr 0 mhr
0.00 mhr 0.00 mhr 0.00 mhr 0.00 vmr_phr 0.00 vmr_phr 0.00 vmr_phr 0 vmr_phr
0.00 vmr_phr 0.00 vmr_phr 0.00 vmr_phr 0.00 vhr_pmr 0.00 vhr_pmr 0.00 vhr_pmr 0 vhr_pmr

Normal distribution:

0 ranking 5 ranking 10 ranking 25 ranking 50 ranking 90 ranking 99 ranking
0.03 vhr 0.02 vhr 0.02 vhr 0.01 vhr 0 vmr 0 vmr 0 vmr
0.02 vmr 0.01 vmr 0.00 vmr 0.00 vmr 0 vhr 0 vhr 0 vhr
0.01 pmr 0.00 pmr 0.00 pmr 0.00 pmr 0 pmr 0 pmr 0 pmr
0.00 phr 0.00 phr 0.00 phr 0.00 phr 0 phr 0 phr 0 phr
0.00 mmr 0.00 mmr 0.00 mmr 0.00 mmr 0 mmr 0 mmr 0 mmr
0.00 mhr 0.00 mhr 0.00 mhr 0.00 mhr 0 mhr 0 mhr 0 mhr
0.00 vmr_phr 0.00 vmr_phr 0.00 vmr_phr 0.00 vmr_phr 0 vmr_phr 0 vmr_phr 0 vmr_phr
0.00 vhr_pmr 0.00 vhr_pmr 0.00 vhr_pmr 0.00 vhr_pmr 0 vhr_pmr 0 vhr_pmr 0 vhr_pmr

MC gains percentiles

Skewed \(t\)-distribution (sstd):

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
0 98.54 95.88 98.09 99.18 99.90 99.85 99.95 99.72
5 98.24 95.55 97.81 99.06 99.87 99.79 99.94 99.64
10 97.91 95.06 97.60 98.89 99.82 99.78 99.92 99.57
25 96.96 93.67 96.73 98.42 99.60 99.68 99.82 99.26
50 94.42 90.66 94.89 97.19 98.89 99.11 99.53 98.35
100 86.53 84.01 88.48 93.76 94.62 96.52 97.40 93.64
200 59.55 64.45 58.53 80.88 63.21 83.14 81.81 67.25
300 30.57 45.25 22.12 62.83 20.52 57.69 51.89 34.04
400 12.38 29.22 4.27 43.97 2.68 32.80 24.25 13.33
500 4.05 17.64 0.42 28.05 0.14 15.30 8.30 3.96
1000 0.00 0.69 0.00 0.90 0.00 0.08 0.01 0.00

Standardized \(t\)-distribution (std):

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
0 99.92 99.93 99.29 99.57 100.00 100.00 100.00 99.98
5 99.91 99.89 99.26 99.48 100.00 100.00 100.00 99.97
10 99.90 99.86 99.16 99.36 100.00 100.00 100.00 99.97
25 99.79 99.77 99.07 98.68 99.97 99.99 99.98 99.95
50 99.44 99.43 98.71 97.08 99.93 99.86 99.89 99.91
100 97.24 97.56 97.82 91.37 99.72 99.15 99.27 99.76
200 85.25 87.19 94.22 71.23 97.47 92.05 90.13 97.86
300 66.73 73.08 87.24 50.68 89.35 73.72 67.15 91.22
400 47.04 57.48 75.62 34.36 73.25 52.35 43.09 77.78
500 32.45 44.15 62.30 22.57 52.86 33.00 24.39 60.20
1000 4.20 9.59 16.84 2.48 6.20 2.30 0.98 9.95

Normal distribution:

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
0 99.98 99.97 99.99 100.00 100.00 100.00 100.00 100.00
5 99.97 99.94 99.99 100.00 100.00 100.00 100.00 100.00
10 99.94 99.91 99.98 99.98 100.00 100.00 100.00 100.00
25 99.82 99.81 99.95 99.95 100.00 100.00 100.00 100.00
50 98.97 99.26 99.66 99.76 99.97 99.99 100.00 100.00
100 93.16 96.31 95.60 98.23 98.98 99.78 99.68 99.44
200 63.54 78.98 58.13 88.14 69.29 93.87 89.96 81.07
300 32.36 56.15 20.52 70.34 22.63 73.77 62.19 42.63
400 13.97 36.76 4.82 50.71 4.22 47.44 33.40 16.63
500 5.63 22.34 0.89 34.42 0.87 26.11 15.81 5.72
1000 0.07 1.64 0.00 4.11 0.00 0.51 0.23 0.07

Best ranking for MC gains percentiles

Skewed \(t\)-distribution (sstd):

0 ranking 5 ranking 10 ranking 25 ranking 50 ranking 100 ranking
99.95 vmr_phr 99.94 vmr_phr 99.92 vmr_phr 99.82 vmr_phr 99.53 vmr_phr 97.40 vmr_phr
99.90 mmr 99.87 mmr 99.82 mmr 99.68 mhr 99.11 mhr 96.52 mhr
99.85 mhr 99.79 mhr 99.78 mhr 99.60 mmr 98.89 mmr 94.62 mmr
99.72 vhr_pmr 99.64 vhr_pmr 99.57 vhr_pmr 99.26 vhr_pmr 98.35 vhr_pmr 93.76 phr
99.18 phr 99.06 phr 98.89 phr 98.42 phr 97.19 phr 93.64 vhr_pmr
98.54 vmr 98.24 vmr 97.91 vmr 96.96 vmr 94.89 pmr 88.48 pmr
98.09 pmr 97.81 pmr 97.60 pmr 96.73 pmr 94.42 vmr 86.53 vmr
95.88 vhr 95.55 vhr 95.06 vhr 93.67 vhr 90.66 vhr 84.01 vhr
200 ranking 300 ranking 400 ranking 500 ranking 1000 ranking
83.14 mhr 62.83 phr 43.97 phr 28.05 phr 0.90 phr
81.81 vmr_phr 57.69 mhr 32.80 mhr 17.64 vhr 0.69 vhr
80.88 phr 51.89 vmr_phr 29.22 vhr 15.30 mhr 0.08 mhr
67.25 vhr_pmr 45.25 vhr 24.25 vmr_phr 8.30 vmr_phr 0.01 vmr_phr
64.45 vhr 34.04 vhr_pmr 13.33 vhr_pmr 4.05 vmr 0.00 vmr
63.21 mmr 30.57 vmr 12.38 vmr 3.96 vhr_pmr 0.00 pmr
59.55 vmr 22.12 pmr 4.27 pmr 0.42 pmr 0.00 mmr
58.53 pmr 20.52 mmr 2.68 mmr 0.14 mmr 0.00 vhr_pmr

Standardized \(t\)-distribution (std):

0 ranking 5 ranking 10 ranking 25 ranking 50 ranking 100 ranking
100.00 mmr 100.00 mmr 100.00 mmr 99.99 mhr 99.93 mmr 99.76 vhr_pmr
100.00 mhr 100.00 mhr 100.00 mhr 99.98 vmr_phr 99.91 vhr_pmr 99.72 mmr
100.00 vmr_phr 100.00 vmr_phr 100.00 vmr_phr 99.97 mmr 99.89 vmr_phr 99.27 vmr_phr
99.98 vhr_pmr 99.97 vhr_pmr 99.97 vhr_pmr 99.95 vhr_pmr 99.86 mhr 99.15 mhr
99.93 vhr 99.91 vmr 99.90 vmr 99.79 vmr 99.44 vmr 97.82 pmr
99.92 vmr 99.89 vhr 99.86 vhr 99.77 vhr 99.43 vhr 97.56 vhr
99.57 phr 99.48 phr 99.36 phr 99.07 pmr 98.71 pmr 97.24 vmr
99.29 pmr 99.26 pmr 99.16 pmr 98.68 phr 97.08 phr 91.37 phr
200 ranking 300 ranking 400 ranking 500 ranking 1000 ranking
97.86 vhr_pmr 91.22 vhr_pmr 77.78 vhr_pmr 62.30 pmr 16.84 pmr
97.47 mmr 89.35 mmr 75.62 pmr 60.20 vhr_pmr 9.95 vhr_pmr
94.22 pmr 87.24 pmr 73.25 mmr 52.86 mmr 9.59 vhr
92.05 mhr 73.72 mhr 57.48 vhr 44.15 vhr 6.20 mmr
90.13 vmr_phr 73.08 vhr 52.35 mhr 33.00 mhr 4.20 vmr
87.19 vhr 67.15 vmr_phr 47.04 vmr 32.45 vmr 2.48 phr
85.25 vmr 66.73 vmr 43.09 vmr_phr 24.39 vmr_phr 2.30 mhr
71.23 phr 50.68 phr 34.36 phr 22.57 phr 0.98 vmr_phr

Normal distribution:

0 ranking 5 ranking 10 ranking 25 ranking 50 ranking 100 ranking
100.00 phr 100.00 phr 100.00 mmr 100.00 mmr 100.00 vmr_phr 99.78 mhr
100.00 mmr 100.00 mmr 100.00 mhr 100.00 mhr 100.00 vhr_pmr 99.68 vmr_phr
100.00 mhr 100.00 mhr 100.00 vmr_phr 100.00 vmr_phr 99.99 mhr 99.44 vhr_pmr
100.00 vmr_phr 100.00 vmr_phr 100.00 vhr_pmr 100.00 vhr_pmr 99.97 mmr 98.98 mmr
100.00 vhr_pmr 100.00 vhr_pmr 99.98 pmr 99.95 pmr 99.76 phr 98.23 phr
99.99 pmr 99.99 pmr 99.98 phr 99.95 phr 99.66 pmr 96.31 vhr
99.98 vmr 99.97 vmr 99.94 vmr 99.82 vmr 99.26 vhr 95.60 pmr
99.97 vhr 99.94 vhr 99.91 vhr 99.81 vhr 98.97 vmr 93.16 vmr
200 ranking 300 ranking 400 ranking 500 ranking 1000 ranking
93.87 mhr 73.77 mhr 50.71 phr 34.42 phr 4.11 phr
89.96 vmr_phr 70.34 phr 47.44 mhr 26.11 mhr 1.64 vhr
88.14 phr 62.19 vmr_phr 36.76 vhr 22.34 vhr 0.51 mhr
81.07 vhr_pmr 56.15 vhr 33.40 vmr_phr 15.81 vmr_phr 0.23 vmr_phr
78.98 vhr 42.63 vhr_pmr 16.63 vhr_pmr 5.72 vhr_pmr 0.07 vmr
69.29 mmr 32.36 vmr 13.97 vmr 5.63 vmr 0.07 vhr_pmr
63.54 vmr 22.63 mmr 4.82 pmr 0.89 pmr 0.00 pmr
58.13 pmr 20.52 pmr 4.22 mmr 0.87 mmr 0.00 mmr

Summary statistics

Fit summary

Summary for fit of log returns to an F-S skew standardized Student-t distribution.
m is the location parameter.
s is the scale parameter.
nu is the estimated degrees of freedom, or shape parameter.
xi is the estimated skewness parameter.

Skewed \(t\)-distribution (sstd):

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
m 0.060 0.065 0.058 0.078 0.052 0.074 0.070 0.066
s 0.101 0.150 0.121 0.113 0.098 0.139 0.119 0.090
nu 3.574 3.144 2.275 3.856 3.032 3.095 2.965 3.569
xi 0.000 0.002 0.477 0.015 0.023 0.006 0.002 0.003
R^2 0.993 0.991 0.991 0.968 0.992 0.979 0.977 0.995

Standardized \(t\)-distribution (std):

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
m 0.084 0.090 0.102 0.073 0.113 0.072 0.067 0.124
s 0.106 0.122 0.345 0.119 0.895 0.117 0.108 0.793
nu 4.844 7.368 2.045 5682540.710 2.006 8971817.739 9916906.918 2.012
R^2 0.935 0.955 0.918 0.923 0.908 0.937 0.924 0.919

Normal distribution:

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
m 0.064 0.077 0.061 0.085 0.063 0.081 0.076 0.069
s 0.081 0.099 0.063 0.101 0.071 0.099 0.090 0.081
R^2 0.933 0.954 0.916 0.923 0.911 0.937 0.924 0.927

AIC and BIC

AIC

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
sstd -28.098 -21.425 -33.230 -23.741 -33.930 -22.790 -26.979 -30.291
std -16.385 -11.623 -22.924 -11.324 -20.406 -11.817 -13.869 -16.796
normal -20.316 -15.218 -27.005 -14.616 -23.809 -15.345 -17.593 -20.613

BIC

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
sstd -25.838 -19.165 -30.970 -21.482 -31.670 -20.530 -24.720 -28.031
std -14.125 -9.363 -20.664 -9.064 -18.146 -9.558 -11.609 -14.536
normal -18.056 -12.958 -24.746 -12.357 -21.550 -13.085 -15.333 -18.353

Kappa

Let \(\{X_{g,i}\}\) be Gaussian distributed with mean \(\mu\) and scale \(\sigma\).

Let \(\{X_{\nu,i}\}\) be \(t\)-distributed, scaled such that \(\mathbb{M}^{\nu}(1) = \mathbb{M}^{g}(1) = \sqrt{\frac{2}{\pi}} \sigma\).

Given \(n_g\), we want to determine and \(n_{\nu}^{*}\) such that

\[\text{Var}\left[\sum_i^{n_g} X_{g,i}\right] = \text{Var}\left[\sum_i^{n_{\nu}^{*}} X_{\nu,i}\right]\]

For iid. r.v \(\{X_i\}\):

\[S_n = X_1 + X_2 + \dots + X_n\] \[\mathbb{M}(n) = \mathbb{E}(\lvert S_n - \mathbb{E}(S_n)\rvert)\] Taleb’s convergence metric (\(\kappa\)):

The “rate” of convergence for \(n\) summands vs \(n_0\), i.e. the improved convergence achieved by \(n - n_0\) additional terms, is given by \(\kappa(n_0, n)\):

\[\kappa(n_0, n) = 2 - \dfrac{\log(n) - \log(n_0)}{\log\left(\frac{\mathbb{M}(n)}{\mathbb{M}(n_0)}\right)}\]

\(\kappa\)

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
0.17 0.21 0.33 0.15 0.22 0.22 0.23 0.17

\(n_{min}\)

What is the minimum value of \(n_{\nu}\), the number of observations from a given skewed \(t\)-distribution, we need to achieve the same degree of convergence as with \(n_g=30\) observations from a Gaussian distribution with the same mean and standard deviation?

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
62 78 164 57 78 78 81 62

Fit statistics ranking

Skewed \(t\)-distribution (sstd):

m ranking s ranking R^2 ranking
0.078 phr 0.090 vhr_pmr 0.995 vhr_pmr
0.074 mhr 0.098 mmr 0.993 vmr
0.070 vmr_phr 0.101 vmr 0.992 mmr
0.066 vhr_pmr 0.113 phr 0.991 vhr
0.065 vhr 0.119 vmr_phr 0.991 pmr
0.060 vmr 0.121 pmr 0.979 mhr
0.058 pmr 0.139 mhr 0.977 vmr_phr
0.052 mmr 0.150 vhr 0.968 phr

Standardized \(t\)-distribution (std):

m ranking s ranking R^2 ranking
0.124 vhr_pmr 0.106 vmr 0.955 vhr
0.113 mmr 0.108 vmr_phr 0.937 mhr
0.102 pmr 0.117 mhr 0.935 vmr
0.090 vhr 0.119 phr 0.924 vmr_phr
0.084 vmr 0.122 vhr 0.923 phr
0.073 phr 0.345 pmr 0.919 vhr_pmr
0.072 mhr 0.793 vhr_pmr 0.918 pmr
0.067 vmr_phr 0.895 mmr 0.908 mmr

Normal distribution:

m ranking s ranking R^2 ranking
0.085 phr 0.063 pmr 0.954 vhr
0.081 mhr 0.071 mmr 0.937 mhr
0.077 vhr 0.081 vhr_pmr 0.933 vmr
0.076 vmr_phr 0.081 vmr 0.927 vhr_pmr
0.069 vhr_pmr 0.090 vmr_phr 0.924 vmr_phr
0.064 vmr 0.099 mhr 0.923 phr
0.063 mmr 0.099 vhr 0.916 pmr
0.061 pmr 0.101 phr 0.911 mmr

Monte Carlo simulations summary

Monte Carlo simulations of portfolio index values (currency values).
Statistics are given for the final state of all paths.
Probability of down-and-out is calculated as the share of paths that reach 0 at some point. All subsequent values for a path are set to 0, if the path reaches at any point.
0 is defined as any value below a threshold.
dai_pct (for down-and-in) is the probability of losing money. This is calculated as the share of paths finishing below index 100.

## Number of paths: 10000

Skewed \(t\)-distribution (sstd):

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
mc_m 365.22 438.43 342.27 534.24 352.73 480.78 449.93 389.07
mc_s 144.99 242.02 113.87 240.41 92.05 171.16 141.79 134.00
mc_min 3.00 3.00 0.04 9.81 76.71 57.75 78.93 9.02
mc_max 1061.35 1733.76 933.98 1791.85 932.38 1698.31 1317.63 1447.45
dao_pct 0.02 0.06 0.12 0.00 0.00 0.00 0.00 0.00
dai_pct 1.36 3.89 1.83 0.66 0.09 0.09 0.03 0.30

Standardized \(t\)-distribution (std):

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
mc_m 594.24 701.71 1.293393e+20 498.88 1439.39 601.20 546.42 4.300181e+05
mc_s 312.33 413.32 1.293393e+22 283.05 54541.36 248.89 211.17 4.192550e+07
mc_min 65.73 78.35 3.000000e-02 57.83 122.67 122.23 127.60 9.973000e+01
mc_max 5603.32 5079.93 1.293393e+24 3108.62 5410256.83 2395.18 2186.83 4.191411e+09
dao_pct 0.00 0.00 9.000000e-02 0.00 0.00 0.00 0.00 0.000000e+00
dai_pct 0.08 0.09 6.900000e-01 0.40 0.00 0.00 0.00 1.000000e-02

Normal distribution:

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
mc_m 387.42 513.18 351.29 610.65 369.91 561.39 501.78 428.70
mc_s 147.82 243.69 100.63 294.23 89.78 187.95 164.02 130.13
mc_min 92.96 82.74 102.84 99.94 131.02 163.84 160.98 158.70
mc_max 1434.40 3114.30 892.69 3527.40 944.67 1814.66 1700.44 2093.70
dao_pct 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
dai_pct 0.01 0.02 0.00 0.01 0.00 0.00 0.00 0.00

Ranking

Skewed \(t\)-distribution (sstd):

mc_m ranking mc_s ranking mc_min ranking mc_max ranking dao_pct ranking dai_pct ranking
534.24 phr 92.05 mmr 78.93 vmr_phr 1791.85 phr 0.00 phr 0.03 vmr_phr
480.78 mhr 113.87 pmr 76.71 mmr 1733.76 vhr 0.00 mmr 0.09 mmr
449.93 vmr_phr 134.00 vhr_pmr 57.75 mhr 1698.31 mhr 0.00 mhr 0.09 mhr
438.43 vhr 141.79 vmr_phr 9.81 phr 1447.45 vhr_pmr 0.00 vmr_phr 0.30 vhr_pmr
389.07 vhr_pmr 144.99 vmr 9.02 vhr_pmr 1317.63 vmr_phr 0.00 vhr_pmr 0.66 phr
365.22 vmr 171.16 mhr 3.00 vmr 1061.35 vmr 0.02 vmr 1.36 vmr
352.73 mmr 240.41 phr 3.00 vhr 933.98 pmr 0.06 vhr 1.83 pmr
342.27 pmr 242.02 vhr 0.04 pmr 932.38 mmr 0.12 pmr 3.89 vhr

Standardized \(t\)-distribution (std):

mc_m ranking mc_s ranking mc_min ranking mc_max ranking dao_pct ranking dai_pct ranking
1.293393e+20 pmr 2.111700e+02 vmr_phr 127.60 vmr_phr 1.293393e+24 pmr 0.00 vmr 0.00 mmr
4.300181e+05 vhr_pmr 2.488900e+02 mhr 122.67 mmr 4.191411e+09 vhr_pmr 0.00 vhr 0.00 mhr
1.439390e+03 mmr 2.830500e+02 phr 122.23 mhr 5.410257e+06 mmr 0.00 phr 0.00 vmr_phr
7.017100e+02 vhr 3.123300e+02 vmr 99.73 vhr_pmr 5.603320e+03 vmr 0.00 mmr 0.01 vhr_pmr
6.012000e+02 mhr 4.133200e+02 vhr 78.35 vhr 5.079930e+03 vhr 0.00 mhr 0.08 vmr
5.942400e+02 vmr 5.454136e+04 mmr 65.73 vmr 3.108620e+03 phr 0.00 vmr_phr 0.09 vhr
5.464200e+02 vmr_phr 4.192550e+07 vhr_pmr 57.83 phr 2.395180e+03 mhr 0.00 vhr_pmr 0.40 phr
4.988800e+02 phr 1.293393e+22 pmr 0.03 pmr 2.186830e+03 vmr_phr 0.09 pmr 0.69 pmr

Normal distribution:

mc_m ranking mc_s ranking mc_min ranking mc_max ranking dao_pct ranking dai_pct ranking
610.65 phr 89.78 mmr 163.84 mhr 3527.40 phr 0 vmr 0.00 pmr
561.39 mhr 100.63 pmr 160.98 vmr_phr 3114.30 vhr 0 vhr 0.00 mmr
513.18 vhr 130.13 vhr_pmr 158.70 vhr_pmr 2093.70 vhr_pmr 0 pmr 0.00 mhr
501.78 vmr_phr 147.82 vmr 131.02 mmr 1814.66 mhr 0 phr 0.00 vmr_phr
428.70 vhr_pmr 164.02 vmr_phr 102.84 pmr 1700.44 vmr_phr 0 mmr 0.00 vhr_pmr
387.42 vmr 187.95 mhr 99.94 phr 1434.40 vmr 0 mhr 0.01 vmr
369.91 mmr 243.69 vhr 92.96 vmr 944.67 mmr 0 vmr_phr 0.01 phr
351.29 pmr 294.23 phr 82.74 vhr 892.69 pmr 0 vhr_pmr 0.02 vhr

Compare Gaussian and skewed t-distribution fits

Gaussian fits

Gaussian QQ plots

Gaussian vs skewed t

Probability in percent that the smallest and largest (respectively) observed return for each fund was generated by a normal distribution:

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
P_norm(X_min) 0.070 0.088 0.389 0.582 0.161 0.255 0.265 0.155
P_norm(X_max) 13.230 11.876 12.922 15.359 15.936 13.198 15.397 15.556
P_t(X_min) 3.743 5.457 3.475 4.567 4.100 5.035 4.182 3.022
P_t(X_max) 0.000 0.001 2.818 0.023 0.052 0.004 0.000 0.001

Average number of years between min or max events (respectively):

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
norm: avg yrs btw min 1.438131e+03 1139.205 256.817 171.880 620.586 392.517 376.706 644.455
norm: avg yrs btw max 7.559000e+00 8.420 7.739 6.511 6.275 7.577 6.495 6.428
t: avg yrs btw min 2.671500e+01 18.324 28.775 21.898 24.387 19.862 23.914 33.089
t: avg yrs btw max 3.834699e+08 178349.076 35.487 4439.617 1930.115 23903.982 236209.123 124926.545

Lilliefors test

p-values for Lilliefors test.
Testing \(H_0\), that log-returns are Gaussian.

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
p value 0.052 0.343 0.024 0.06 0.041 0.251 0.113 0.183

Wittgenstein’s Ruler

For different given probabilities that returns are Gaussian, what is the probability that the distribution is Gaussian rather than skewed t-distributed, given the smallest/largest observed log-returns?

Conditional probabilities for smallest observed log-returns:

Use \(1 - \text{p-value}\) from Lilliefors test as prior probability that the distribution is Gaussian.
\(x_{\text{obs}} = \min(x)\) and \(P[\text{Event}\ |\ \text{Gaussian}] = P_{\text{Gauss}}[X \leq x_{\text{min}}]\):

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
Lillie p-val 0.052 0.343 0.024 0.060 0.041 0.251 0.113 0.183
Prior prob 0.948 0.657 0.976 0.940 0.959 0.749 0.887 0.817
P[Gauss | Event] 0.737 0.210 0.855 0.852 0.730 0.383 0.649 0.448

Use \(1 - \text{p-value}\) from Lilliefors test as prior probability that the distribution is Gaussian.
\(x_{\text{obs}} = \max(x)\) and \(P[\text{Event}\ |\ \text{Gaussian}] = P_{\text{Gauss}}[X \geq x_{\text{max}}]\):

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
Lillie p-val 0.052 0.343 0.024 0.06 0.041 0.251 0.113 0.183
Prior prob 0.948 0.657 0.976 0.94 0.959 0.749 0.887 0.817
P[Gauss | Event] 1.000 1.000 0.995 1.00 1.000 1.000 1.000 1.000

Velliv medium risk (vmr), 2011 - 2023

QQ Plot

Skewed \(t\)-distribution (sstd):

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

Monte Carlo

Sorted portfolio index values for last period of all runs

Convergence

Max vs sum

Max vs sum plots for the first four moments:

MC

IS

Skewed \(t\)-distribution with a normal proposal distribution.

Parameters

## [1] 1.4145605 0.3807834

Objective function plots

Velliv high risk (vhr), 2011 - 2023

QQ Plot

Skewed \(t\)-distribution (sstd):

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

Monte Carlo

Sorted portfolio index values for last period of all runs

Convergence

Max vs sum

Max vs sum plots for the first four moments:

MC

IS

Skewed \(t\)-distribution with a normal proposal distribution.

Parameters

## [1] 1.7391222 0.4858909

Objective function plots

PFA medium risk (pmr), 2011 - 2023

QQ Plot

Skewed \(t\)-distribution (sstd):

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

Monte Carlo

pmr has the sstd fit with the lowest value of nu. Compare with other distributions:

Sorted portfolio index values for last period of all runs

Convergence

Max vs sum

Max vs sum plots for the first four moments:

MC

IS

Skewed \(t\)-distribution with a normal proposal distribution.

Parameters

## [1] 1.3304634 0.2764028

Objective function plots

PFA high risk (phr), 2011 - 2023

QQ Plot

Skewed \(t\)-distribution (sstd):

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

Monte Carlo

phr has the sstd fit with the highest sstd fit with thevalue of nu. Compare with other distributions:

Sorted portfolio index values for last period of all runs

Convergence

Max vs sum

Max vs sum plots for the first four moments:

MC

IS

Skewed \(t\)-distribution with a normal proposal distribution.

Parameters

## [1] 2.0162301 0.4463226

Objective function plots

Mix medium risk (mmr), 2011 - 2023

QQ Plot

Skewed \(t\)-distribution (sstd):

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

Monte Carlo

Sorted portfolio index values for last period of all runs

Convergence

Max vs sum

Max vs sum plots for the first four moments:

MC

IS

Skewed \(t\)-distribution with a normal proposal distribution.

Parameters

## [1] 1.3516393 0.2503782

Objective function plots

Mix high risk (mhr), 2011 - 2023

QQ Plot

Skewed \(t\)-distribution (sstd):

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

Monte Carlo

Sorted portfolio index values for last period of all runs

Convergence

Max vs sum

Max vs sum plots for the first four moments:

MC

IS

Skewed \(t\)-distribution with a normal proposal distribution.

Parameters

## [1] 1.8775189 0.3400818

Objective function plots

Mix vmr+phr (vm_ph), 2011 - 2023

QQ Plot

Skewed \(t\)-distribution (sstd):

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

Monte Carlo

Sorted portfolio index values for last period of all runs

Convergence

Max vs sum

Max vs sum plots for the first four moments:

MC

IS

Skewed \(t\)-distribution with a normal proposal distribution.

Parameters

## [1] 1.7507161 0.3263777

Objective function plots

Mix vhr+pmr (mh_pm), 2011 - 2023

QQ Plot

Skewed \(t\)-distribution (sstd):

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

Monte Carlo

Sorted portfolio index values for last period of all runs

Convergence

Max vs sum

Max vs sum plots for the first four moments:

MC

IS

Skewed \(t\)-distribution with a normal proposal distribution.

Parameters

## [1] 1.540135 0.320090

Objective function plots

Velliv medium risk (vmr), 2011 - 2023

QQ Plot

Skewed \(t\)-distribution (sstd):

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

Monte Carlo

Sorted portfolio index values for last period of all runs

Convergence

Max vs sum

Max vs sum plots for the first four moments:

MC

IS

Skewed \(t\)-distribution with a normal proposal distribution.

Parameters

## [1] 1.4145605 0.3807834

Objective function plots

Appendix

Infinite variance

Taleb, Statistical Consequences Of Fat Tails, p. 97:
“the variance of a finite variance random variable with tail exponent \(< 4\) will be infinite”.

And p. 363:
“The hedging errors for an option portfolio (under a daily revision regime) over 3000 days, under a constant volatility Student T with tail exponent \(\alpha = 3\). Technically the errors should not converge in finite time as their distribution has infinite variance.”

QQ lines

Note: QQ lines by design pass through 1st and 3rd quantiles. They are not trendlines in the sense of linear regression.

Arithmetic vs geometric mean

Let \(m\) be the number of steps in each path and \(n\) be the number of paths. \(a\) is the initial capital. Use arithmetic mean for mean of all paths at time \(t\): \[\dfrac{a (e^{z_1} + e^{z_2} + \dots + e^{z_n})}{n}\] where \[z_i := x_{i, 1} + x_{i, 2} + \dots + x_{i, m}\] Use geometric mean for mean of all steps in a single path \(i\): \[a e^{\frac{x_{i, 1} + x_{i, 2} + \dots + x_{i, m}}{m}} = a \sqrt[m]{e^{x_{i, 1} + x_{i, 2} + \dots + x_{i, m}}}\]

So for Monte Carlo of returns after \(m\) periods, we

  • fit a skewed t-distribution to log-returns and use that distribution to simulate \(\{x_{i, j}\}_j^m\),
  • for each path \(i\), calculate \(100\cdot e^{z_i}\),
  • calculate the mean of \(\{z_i\}_i^n\):
    • \[\bar{z} = 100\dfrac{e^{z_1} + e^{z_2} + \dots + e^{z_n}}{n}\]

For Importance Sampling, we

  • model log-returns on a skewed t-distribution,
  • for each path \(i\), calculate \(100\cdot e^{z_i}\),
  • fit a skewed t-distribution to \(\{z_i\}_i^n\) and use it as our \(f\) density function from which we simulate \(\{h_i\}_i^n\),
    • In our case \(h\) and \(z\) are identical, because we have an idea for a distribution to simulate \(z\), but in general for IS \(h\) could be a function of \(z\).
  • calculate \(w* = \frac{f}{g^*}\), where \(g*\) is our proposal distribution, which minimizes the variance of \(h\cdot w\).
  • calculate the arithmetic mean of \(\{h_i w_i^{*}\}_i^n\):
    • \[100 \dfrac{e^{h_1 w_1^{*}} + e^{h_2 w_2^{*}} + \dots + e^{h_n w_n^{*}}}{n}\]

Average of returns vs returns of average

Math

\[\text{Avg. of returns} := \dfrac{ \left(\dfrac{x_t}{x_{t-1}} + \dfrac{y_t}{y_{t-1}} \right) }{2}\] \[\text{Returns of avg.} := \left(\dfrac{ x_t + y_t }{2}\right) \Big/ \left(\dfrac{ x_{t-1} + y_{t-1} }{2}\right) \equiv \dfrac{ x_t + y_t }{ x_{t-1} + y_{t-1}}\]

For which \(x_1\) and \(y_1\) are \(\text{Avg. of returns} = \text{Returns of avg.}\)?

\[\dfrac{ \left(\dfrac{x_t}{x_{t-1}} + \dfrac{y_t}{y_{t-1}} \right) }{2} = \dfrac{ x_t + y_t }{ x_{t-1} + y_{t-1}}\]

\[\dfrac{x_t}{x_{t-1}} + \dfrac{y_t}{y_{t-1}} = 2 \dfrac{ x_t + y_t }{ x_{t-1} + y_{t-1}}\]

\[(x_{t-1} + y_{t-1}) x_t y_{t-1} + (x_{t-1} + y_{t-1}) x_{t-1} y_t = 2 (x_{t-1}y_{t-1}x_t + x_{t-1}y_{t-1}y_t)\]

\[(x_{t-1}x_ty_{t-1} + y_{t-1}x_ty_{t-1}) + (x_{t-1}x_{t-1}y_t + x_{t-1}y_{t-1}y_t) = 2(x_{t-1}y_{t-1}x_t + x_{t-1}y_{t-1}y_t)\] This is not generally true, but true if for instance \(x_{t-1} = y_{t-1}\).

Example

Definition: R = 1+r

## Let x_0 be 100.
## Let y_0 be 200.
## So the initial value of the pf is 300 .
## Let R_x be 0.5.
## Let R_y be 1.5.

Then,

## x_1 is R_x * x_0 = 50.
## y_1 is R_y * y_0 = 300.

Average of returns:

## 0.5 * (R_x + R_y) = 1

So here the value of the pf at t=1 should be unchanged from t=0:

## (x_0 + y_0) * 0.5 * (R_x + R_y) = 300

But this is clearly not the case:

## 0.5 * (x_1 + y_1) = 0.5 * (R_x * x_0 + R_y * y_0) = 175

Therefore we should take returns of average, not average of returns!

Let’s take the average of log returns instead:

## 0.5 * (log(R_x) + log(R_y)) = -0.143841

We now get:

## (x_0 + y_0) * exp(0.5 * (log(Rx) + log(Ry))) = 259.8076

So taking the average of log returns doesn’t work either.

Simulation of mix vs mix of simulations

Test if a simulation of a mix (average) of two returns series has the same distribution as a mix of two simulated returns series.

## m(data_x): -0.0009009278 
## s(data_x): 0.341505 
## m(data_y): 9.986828 
## s(data_y): 3.86378 
## 
## m(data_x + data_y): 4.992964 
## s(data_x + data_y): 1.964015

m and s of final state of all paths.
_a is mix of simulated returns.
_b is simulated mixed returns.

m_a m_b s_a s_b
99.758 100.039 8.818 9.082
99.733 99.617 8.435 9.034
99.949 99.981 8.728 8.749
100.126 99.525 8.654 8.931
99.998 99.590 8.969 9.095
99.612 99.483 8.891 8.925
99.790 99.733 8.949 8.758
99.847 100.170 8.217 8.806
99.512 100.107 8.719 8.905
99.420 99.888 8.670 8.907 
##       m_a              m_b              s_a             s_b       
##  Min.   : 99.42   Min.   : 99.48   Min.   :8.217   Min.   :8.749  
##  1st Qu.: 99.64   1st Qu.: 99.60   1st Qu.:8.658   1st Qu.:8.830  
##  Median : 99.77   Median : 99.81   Median :8.724   Median :8.916  
##  Mean   : 99.77   Mean   : 99.81   Mean   :8.705   Mean   :8.919  
##  3rd Qu.: 99.92   3rd Qu.:100.02   3rd Qu.:8.873   3rd Qu.:9.008  
##  Max.   :100.13   Max.   :100.17   Max.   :8.969   Max.   :9.095

_a and _b are very close to equal.
We attribute the differences to differences in estimating the distributions in version a and b.

The final state is independent of the order of the preceding steps:

So does the order of the steps in the two processes matter, when mixing simulated returns?

The order of steps in the individual paths do not matter, because the mix of simulated paths is a sum of a sum, so the order of terms doesn’t affect the sum. If there is variation it is because the sets preceding steps are not the same. For instance, the steps between step 1 and 60 in the plot above are not the same for the two lines.

Recall, \[\text{Var}(aX+bY) = a^2 \text{Var}(X) + b^2 \text{Var}(Y) + 2ab \text{Cov}(a, b)\]

var(0.5 * vhr + 0.5 * phr)
## [1] 0.01055146
0.5^2 * var(vhr) + 0.5^2 * var(phr) + 2 * 0.5 * 0.5 * cov(vhr, phr)
## [1] 0.01055146

Our distribution estimate is based on 13 observations. Is that enough for a robust estimate? What if we suddenly hit a year like 2008? How would that affect our estimate?
Let’s try to include the Velliv data from 2007-2010.
We do this by sampling 13 observations from vmrl.

##        m                 s          
##  Min.   :0.05912   Min.   :0.05196  
##  1st Qu.:0.06574   1st Qu.:0.06081  
##  Median :0.06785   Median :0.06831  
##  Mean   :0.06880   Mean   :0.06911  
##  3rd Qu.:0.07076   3rd Qu.:0.07634  
##  Max.   :0.08354   Max.   :0.09154

The meaning of xi

The fit for mhr has the highest xi value of all. This suggests right-skew:

Max vs sum plot

If the Law Of Large Numbers holds true, \[\dfrac{\max (X_1^p, ..., X^p)}{\sum_{i=1}^n X_i^p} \rightarrow 0\] for \(n \rightarrow \infty\).

If not, \(X\) doesn’t have a \(p\)’th moment.

See Taleb: The Statistical Consequences Of Fat Tails, p. 192